1
BLISS: A Blind Spectrum Separation Approach forJamming-Resistant Communications
Srikanth Pagadarai
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute, Worcester, MA, USA
Email:[email protected]
Research Advisor: Professor Alexander M. Wyglinski
Abstract—In this presentation, we discuss some priliminaryresults pertaining to an ongoing project on the developmentofan adaptive signal processing solution combining antenna subsetselection, spectral subtraction, and blind source separation inorder to mitigate the impact of both wideband jamming andco-site interference by extracting individual transmissions frommultiple intercepted mixtures of wireless signals.
I. I NTRODUCTION
With the increasing volume of wireless traffic that theatreoperations require, the probability of transmissions interferingwith each other is steadily growing to the point that new tech-niques need to be employed. Furthermore, to combat remotelyoperated improvised explosive devices, many ground convoystransmit high-power broadband jamming signals, which blocksboth hostile as well as friendly communications. We aim todevise, implement, and evaluate an adaptive signal processingsoftware solution for mitigating the effects of both intentionaland unintentional jamming (including wideband jamming) viathe combination of antenna subset selection, spectral subtrac-tion, and blind source separation (BSS) techniques in orderto extract specific transmissions from a mixture of interceptedwireless signals. The goal of our proposed solution, calledBLInd Spectrum Separation (BLISS), is to enable reliable,high throughput, and robust end-to-end wireless communi-cations in the support of all Department of Navy missions,especially high capacity multimedia (voice, data, imagery)transmissions.
II. T ECHNICAL APPROACH
The BLISS solution (see Figure 1) integrates three well-known adaptive signal processing algorithms found in theopen literature, namely: antenna subset selection, spectralsubtraction, and blind source separation (BSS). Each of thesealgorithms is employed within the BLISS framework in orderto enable the process of extracting individual transmissionsintercepted from several mixtures of wireless signals. Al-though BSS can readily extract transmissions under ideal con-ditions, the BLISS framework will be deployed in challengingoperational scenarios which could significantly degrade theperformance of the communication system, or even cause itto fail. Hence, the other two algorithms, i.e., antenna subsetselection and spectral subtraction, are employed to make the
Fig. 1. Block diagram of the BLISS framework.
BSS process more robust in challenging operational condi-tions. Once the implementation of each of these techniques isperformed, the next step is their integration into a seamlessBLISS framework. Finally, validation of the novel frameworkis performed via computer simulations over a range of differentoperating scenarios.
Blind Signal Separation
Blind signal separation is the task of separating signals whenonly their mixtures are observed. The process is often termed“blind”, with the understanding that both source signals andmixing procedure are unknown. The mathematical model thatdescribes the observed mixtures as a function of independentcomponents is as follows:
x = Hs + n (1)
In reality, the specific mixing model is the paramount pieceof prior information required, and in many scenarios evenknowledge of certain source statistics is necessary. Underthissetting, the channel may generally be construed as a lineartime-invariant (LTI) system, though there is some activityoccurring with nonlinear mixing models. Algorithms whichrely on this concept, the separation-independence equivalence,may be classed as those performing Independent Compo-nent Analysis (ICA). The problem of BSS is then reducedto a mathematical optimization problem, of which severalapproaches exist, e.g., kurtosis, mutual information, crosspower-spectra, entropy, log-likelihood [1]. For instance, theoptimization problem based on kurtosis in two-dimensions can
2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−1
−0.5
0
0.5
1
t
s 1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−1
−0.5
0
0.5
1
t
s 2
(a) Original sources.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−2
−1
0
1
2
t
s1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
−2
−1
0
1
2
t
s2
(b) Estimated sources.
Fig. 2. An illustration of BSS.
be formulated as the minimization of
F(q) = q4
1+ q
4
2(2)
over q2
1+ q
2
2= 1 where the vector,q is a function of the
mixing matrix. The BSS problem for the case of an idealized2× 2 BPSK system is illustrated in Figure 2.
In this work, we will devise an unmixing system designedto separate wireless transmissions operating in the presence ofjamming fields, especially those with wideband characteristics.Furthermore, we will investigate technical challenges that willbe potentially encountered by the BLISS-enabled platform anddevise solutions for them.
Antenna Subset Selection
Multiple-input multiple-output (MIMO) wireless solutionsi.e, the use of multiple antennas at transmitter and receiver,has emerged as a cost-effective technology in making highdata rates a reality. This is due to the inefficiency associatedwith the implementation of a brute force approach of achievinghigh data rates using a single transmit-single receive antennasystem which is theoretically feasible provided that the productof the bandwidth (Hertz) and spectral efficiency (bits persecond per Hertz) matches the desired data rate. However,deploying transceivers with multiple antennas requires mul-tiple RF chains that are typically very expensive. Therefore,there is considerable incentive for low-cost, low-complexity
techniques with the benefits of multiple antennas. Optimalantenna subset selection is one such technique. A selectionof antenna elements, which are typically much cheaper thanRF chains, is made available at the transmitter and/or receiver.Transmission/reception is performed through the optimal sub-set. Several citerion have been proposed in order to arrive atthe optimal subset under a variety of operating conditions [2]–[5].
Spectral Subtraction
Spectral subtraction is a technique which has been tradition-ally employed for enhancing speech corrupted by broadbandnoise. In the context of speech enhancement, the techniqueinvolves subtracting an estimate of the noise power spectrumfrom the speech power spectrum, setting negative differencesto zero, recombining the new power spectrum with the originalphase, and then reconstructing the time waveform [6]–[8].In our implementation of the spectral subtraction, insteadofsubtracting noise, we subtract the power spectral densities ofknown signals from those that need further processing.
Integrated Framework
As for the overall framework that combines antenna subsetselection, spectral subtraction and blind signal separation, byapplying an optimal antenna subset selection algorithm, thesignal at the end of the transmitter passes through the RFchain corresponding to a subset of the available antennas andexperiences the distortion caused by the wireless channel.Thiswireless channel is modeled as a mixing matrix as shown inFigure 2. At the receiver end, a spectral subtraction techniqueis applied to subtract all of the known signals. After the knownsignals are subtracted, the problem is essentially a blind signalseparation problem. We would have a set of N signals that arereceived over M sensors. Nongaussianity of noise is assumedand by applying a BLISS technique especially tailored tothe case of jamming-resistant communications, the signalsarereceived.
REFERENCES
[1] A. Hyvarinen, J. Karhunen, and E. Oja”,Independent Component Anal-ysis. John Wiley & Sons Inc., 2001.
[2] R. U. Nabar, D. A. Gore, and A. J. Paulraj”, “Optimal selection anduse of transmit antennas in wireless systems,” inProc. ICT, (Acapulco,Mexico), 2000.
[3] D. A. Gore, R. U. Nabar, and A. J. Paulraj”, “Selecting an optimal set oftransmit antennas for a lowrank matrix channel,” inProc. IEEE ICASSP,(Istanbul, Turkey), pp. 2785–2788, 2000.
[4] A. Molisch, M. Win, and J. Winters”, “Capacity of MIMO systems withantenna selection,” inProc. IEEE Int. Contr. Conf., (Helsinki, Finland,2001), pp. 570–574, 2001.
[5] D. A. Gore and A. J. Paulraj”, “Space-time block coding with opti-mal antenna selection,” inProc. IEEE ICASSP, (Salt Lake City, UT),p. 24412444, 2001.
[6] S. A. M. Manzuri, R. Dianat, and J. Kabudian”, “An improved spectralsubtraction speech enhancement system by using an adaptivespectralestimator,” in Proc. IEEE Canadian Conf. on Elec. and Comp. Engg.,May 2005.
[7] R. Singh and P. Rao”, “Spectral subtraction speech enhancement withrasta filtering,” inProc. of National Conf. on Commns., (Kanpur, India),2007.
[8] Y. Nomura, J. Lu, H. Sekiya, and T. Yahagi”, “Spectral subtraction basedon speech/noise-dominant classification,” inProc. IEEE Int. Workshop onAcoustics, Echo and Noise Control, pp. 127–130, 2003.
1
System Modeling, Simulation, and Design forBeamforming Using Simulink HDL Coder
Yanjie PengDepartment of Electrical and Computer Engineering
Worcester Polytechnic Institute, Worcester, MA, USAEmail: [email protected]
Research Advisor: Professor Xinming Huang and Professor Andrew G. Klein
Abstract—Beamforming is a signal processing technique usedin sensor arrays for directional signal transmission or reception.It has found numerous applications in radar, sonar, seismology,wireless communications, speech, acoustics, and biomedicine.Adaptive beamforming is used to detect and estimate the signal-of-interest at the output of a sensor array by means of data-adaptive spatial filtering and interference rejection. We focus onthe study of using Matlab/Simulink tools for model simulationand hardware design of the traditional linear array beamformingusing QR-RLS (QR-decomposition-based recursive least-squares)algorithm. We present the details on the theoretical algorithm,corresponding simulation model, and the FPGA result of thehardware generated by Simulink HDL Coder.
I. INTRODUCTION
In the recent decade, several technologies have been pro-posed to achieve MIMO (multiple-input and multiple-output)from a conventional SISO (single-input and single-output)system. Among them beamforming [1] is a signal processingtechnique used in sensor arrays for directional signal trans-mission or reception. It has found numerous applications inradar, sonar, seismology, wireless communications, speech,acoustics, and biomedicine. Adaptive beamforming is used todetect and estimate the signal-of-interest at the output of asensor array by means of data-adaptive spatial filtering andinterference rejection.
We present the design of the traditional linear array beam-forming based on QR-RLS (QR-decomposition-based recur-sive least-squares) algorithm [1] using Simulink HDL Coder[5]. The theoretical algorithm is discussed in Section II.System modeling using Simulink HDL Coder is introducedin Section III, followed by hardware implementation result inSection IV. Conclusions and future work are given in SectionV.
II. LINEAR ARRAY ADAPTIVE BEAMFORMING
The block diagram of the traditional linear array beamform-ing (M sensors) is presented in Fig. 1. s(φ) is the steeringvector containing the information of the direction of the targetsignal. w∗1(n), w∗2(n), . . . w∗M (n) is the adjustable complexweight for sensor 1,2,...M , respectively.
The requirements for MVDR (minimum-variance distortion-less response) beamforming is: 1) protecting the target signal,
This work is supported by The Mathworks, Inc., Natick , Massachusetts01760, USA
...
Adaptive Weight Computation
Array of sensors
Output
Adjustable complex weights
Steering vector ( )φs
*
1( )w n
...
...
2
* ( )w n
* ( )M
w n
...
Figure 1. Beamforming
and 2) minimizing the interference. The QR-RLS algorithmis a numerically stable solution to MVDR problem. Thealgorithm is shown in Algorithm 1. The output of beamformeris e(n) = e′(n)
||a(n)||2 .
III. SIMULATION MODEL
The systolic array structure [3] is very suitable for im-plementing the QR-RLS algorithm. The simulation model ofsystolic array for 3-sensor beamforming build by SimulinkHDL Coder is shown in Figure 2. The array contains twotypes of processing cells: boundary cells (blue) and internalcells (red). The boundary cells perform the vectoring operationon received signal to form rotation angles used by internalcells. The internal cells perform Givens rotations of thereceived signal by the angles passed from the boundary cells.Simulation result given 50 snapshots is shown in Fig. 3.
IV. HARDWARE IMPLEMENTATION RESULTS
After fixed point simulation, we can obtain synthesizableverilog and/or hdl code directly by HDL Coder. The FPGAresult is shown in Table 2.
V. CONCLUSIONS AND FUTURE WORK
We present the design of the traditional linear array beam-forming based on QR-RLS (QR-decomposition-based recur-sive least-squares) algorithm using Simulink HDL Coder. The
2
Algorithm 1 QR-RLS Algorithm
• Initial condition Φ1/2(0) = I, and a(n) = s(φ)• For n = 1, 2, . . ., compute Φ1/2(n− 1) u(n)
aH(n− 1) 00T 1
Θ(n) =
Φ1/2(n) 0aH(n) −e′(n)γ−1/2(n)
uH(n)ΦH/2(n− 1) γ1/2(n)
where s(φ) = [1, e−jφ, ..., e−j(M−1)φ]T is the steering vector, u(n) is the received data at time n, and Θ(n) denotes Givens rotation
Output10
aH39
aH28
aH17
phiSqrt326
phiSqrt315
phiSqrt214
phiSqrt333
phiSqrt222
phiSqrt111
0
cell43
uIn
cIn
sIn
uOut
xOut
cell42
uIn
cIn
sIn
uOut
xOut
cell41
uIn
cIn
sIn
uOut
xOut
cell33
uIn
gIn
c s gOut
xOut
cell32uI
n
cIn
sIn
cOut
sOut
uOut
xOut
cell31
uIn
cIn
sIn
cOut
sOut
uOut
xOut
cell22
uIn
gIn
c s gOut
xOut
cell21
uIn
cIn
sIn
cOut
sOut
uOut
xOut
cell11
uIn
gIn
c s gOut
xOut
1/z1/z
1/z
1
uIn33
uIn22
uIn11
Figure 2. Systolic array for 3-sensor beamforming
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−60
−50
−40
−30
−20
−10
0
10
sinθ
Am
plitu
de r
espo
nse
(dB
)
MVDR Beamforming (3 sensors, 50 snapshots)
Targe Angle = sin−1(0.2)Interference Angle = sin−1(0)
Interference−to−noise ratio = 20 dbInterference−to−noise ratio = 30 dbInterference−to−noise ratio = 40 db
Figure 3. Simulation result of 3-sensor beamforming
Product Version Xilinx ISE V11.2
Target Device Xilinx Virtex5 XC5VLX330
Number of Slice Registers 1,824 out of 207,360 (1%)
Number of Slice LUTs 5,355 out of 207,360 (2%)
Number of DSP48Es 103 out of 192 (53%)
Maximum Speed 72.6 MHz
Table IFPGA RESULT FOR 3-SENSOR BEAMFORMING
theoretical algorithm, system modeling and simulation, andhardware implementation results are discussed.
In Section II boundary cells require implementations ofsquare root (sqrt) and reciprocal function. One well-knownand relatively simple method for computing sqrt and reciprocalis the CORDIC (Coordinate Rotation Digital Computer) [4]algorithm. The elemental operations required in the CORDICalgorithm are addition, subtraction, bit-shift, and table look-up.All of these functions are efficiently supported by FPGA archi-tectures, so the CORDIC algorithm is a good candidate for theboundary cells implementation. In [4], two pipelined CORDICengines are in use in the boundary cell. Since CORDICimplementation of sqrt and reciprocal is not supported so far inSimulink HDL coder, we used the Newton–Raphson methodin our design which would consume significant DSP resource,but also constrain the circuit speed. We will develop CORDICblock for sqrt and reciprocal implementation in the future.
REFERENCES
[1] S. Haykin, “Adaptive filter theory,” Prentice Hall, 4th edition, 2002.[2] J. G. McWhirter,“Recursive least-squares minimization using a systolic
array,” in Proc. SPIE, Real-Time Signal Processing VI, vol.431, pp.105-112, San Diego, CA, 1983.
[3] J. E. Volder, “The CORDIC trigonometric computing technique,” IRETrans. on Electronic Computers, pp. 330-334, Sep. 1959.
[4] C. Dick, F. Harris, M. Pajic, and D. Vuletic, “Real-time QRD-basedbeamforming on an FPGA platform,” in Proc. the 40th Asilomar Conf.on Signals, Systems and Computers, pp. 1200-1204, Oct, 2006.
[5] Simulink HDL Coder 1.6, The Mathworks, Inc.http://www.mathworks.com/products/slhdlcoder/
Wind Energy Conversion System Operation asPower Generator and Active Filter
Grazia Todeschini
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute, Worcester, MA, USA
Email: [email protected]
Research Advisor: Professor Alexander E. Emanuel
Abstract—This research deals with the performance of aWind Energy Conversion System (WECS) operating as powergenerator and Active Filter simultaneously. As a power generator,the WECS converts wind energy into electric energy; as an AF,it sinks the harmonic currents injected by Non-Linear Loads(NLLs) connected at the same feeder.
Three control systems are developed to ensure the describedoperation and a specific study regarding the compensation of thetriplen harmonics is carried out. WECS derating and voltagedistortion are expressed as function of the harmonic currentsinjected by the system.
The WECS performance as generator and AF has been studiedfor steady-state analysis, fast transients (voltage variations)andslow transients (wind speed variations). Simulation results ofa typical plant show that the proposed control systems allowoperating the WECS as power generator and AF both duringsteady state and transient operation; the described operationcauses power loss increase and voltage distortion that require aconservative choice of the WECS components and require systemderating.
I. I NTRODUCTION
Distributed generation represents an answer to the growingdemand of electric energy and to the increasing environmentalconcerns. Among alternative sources, wind energy is one of themost promising technologies. In the developed countries, windcapacity has been growing at a rate of 20% to 30% a year overthe last decade [1]. The rapid growth of the wind industry isdue to different factors, including [2]: supporting governmentpolicies, advances in wind power technology that causedcost reduction and performance improvement, environmentalconcerns and the deregulation process.
Given the advances in power electronics and control, ap-plications such as reactive power compensation, static transferswitches, energy storage, variable-speed generation, voltagecontrol and dynamic reactive power support are commonlyfound in modern wind power plants [3]: these applications areknown as ‘ancillary services’. The present research presentedin this paper investigate one of the ancillary services: activefilter operation.
II. T HE STUDIED SYSTEM
The studied system is shown in Fig. 1. The Doubly-FedInduction Generator (DFIG) stator terminals are connectedto
the Point of Common Coupling (PCC) through a transformerand a feeder, represented by the equivalent resistance andinductanceRc and Lc. The feeder that connects the Non-Linear Load (NLL) to the PCC is represented by the equivalentresistance and inductanceRh andLh. The DFIG rotor is sup-plied by two back-to-back connected converters: the rotor sideconverter (RSC) and the line side converter (LSC). The feederthat connects the LSC to the PCC has the equivalent resistanceand inductanceRL andLL. The RSC and LSC power switchesare driven by means of Pulse Width Modulation (PWM).The control system design is performed in an equivalentdq0
domain obtained by applying Park transformation to the three-phase variables [4].
III. C OMPENSATION BY MEANS OF COMBINED
MODULATION
As proved in [5], the most effective AF strategy uses bothpower converters and the DFIG by combining compensationby means of RSC and LSC modulation. This strategy is named‘combined modulation’ (CM). The aim of this method is todistribute the harmonic power flow and therefore the powerloss increase among the DFIG, the RSC and the LSC.
When compensation by means of CM is applied, the zero-sequence harmonics injection is obtained by modulating theLSC, and compensation of the symmetrical harmonic compo-nents is obtained by modulating the RSC. In terms ofdq0
components, the reference harmonic currents for the LSC andthe RSC control systems are as follows:
idL,ref = −i0h (1)
ids,ref = −idh (2)
iqs,ref = −iqh (3)
whereidh, iqh, i0h are thedq0 currents measured at the NLLterminals.
A. Steady-state analysis
Steady-state analysis allows verifying the validity of theproposed control system and quantifiyng its effects.
The current and voltage THD (Total Harmonic Distortion)are measured at the PCC for a case study, before and after
Fig. 1. WECS (Wind Energy Conversion System) configuration: the Doubly-Fed Induction Generator (DFIG) stator is connected to the PCC through a line;the DFIG rotor is supplied by two back-to-back connected converters; the dc-link central tap is tied to the neutral. A NLLis connected to the PCC.
TABLE ICURRENT AND VOLTAGE THD AT THE PCCFOR THE CASE STUDY: COMPARISON BETWEEN THE THREE COMPENSATION METHODS.
No Compensation With compensation Limit USRSC mod. LSC mod. Combined mod.
(a) Current THD 25% 16% 7.5% 7.0% 5-20%
(b) Voltage THD 115 4.7% 5.5% 4.0% 5%
harmonic compensation is implemented. Tab. I lists the re-sults and shows a significant THD reduction when harmoniccompensation is applied.
Two effects of the proposed application on the WECSoperation are also identified:
1) power loss increase and consequent temperature rise; un-der certain operating conditions, derating of the WECSis necessary to limit the power loss and the devicestemperature.
2) voltage distortion due to the harmonic current flowand harmonic voltage drop on the line impedances.It results that the the measured peak voltage at thestator terminals is above the rated value and requiresconservative insulation design.
B. Transient analysis
The system transient response is investigated for two typesof disturbances: voltage variation (fast transients) and windspeed variation (slow transients). An extensive number ofcases have been studied and the results are summarized asfollows:
• harmonic compensation is not affected by transient oper-ation, since the reference harmonic currents are measuredat the NLL terminals;
• derating helps reducing severity of current during thetransients because it causes a reduction of the fundamen-tal current amplitude;
• LVRT ability is verified: the WECS continues to provide
power and is not damaged according to the requirementspresented in [6];
• wind speed variation introduces low frequency currentdisturbances, due to the mechanical system response.
IV. CONCLUSION
The simulation results show that the WECS can be usedsimultaneously as Active Filter and Power Generator. Steady-state analysis shows that to safely operate the system athigh wind speeds, derating and a conservative choice of thesystem components are necessary. Transient phenomena haveno effects on the WECS ability to inject harmonic currents.
REFERENCES
[1] Z. Chen and F. Blaabjerg, “Wind energy: The world’s fastest growingenergy source,”IEEE Power Electronics Society Newsletter, vol. 18, no.3, pp. 15–19, 2006.
[2] T. Ackermann, Wind Power in Power Systems, John Wiley and Sons,Hoboken, NJ 07030-5774, US, 2005.
[3] Wachtel S., J. Marques, E. Quitman, and M. Schellschmidt, “Wind energyconverters with FACTS capabilities and the benefits for the integrationof wind power plants into power systems,” inProc. European WindEnergy Conference and Exhibition (EWEC), Milan, 4, May 7-10 2007,pp. 1761–65.
[4] B. K. Bose, Power Electronics and AC drives, Prentice-Hall, UpperSaddle River, NJ, 07458, US, 2001.
[5] G. Todeschini and A. E. Emanuel, “Wind energy conversion system asan active filter: Design and comparison of three control systems (part I),”Submitted to IET, Companion paper.
[6] FERC Order No. 661A,Interconnection for Wind Energy, 2005.
1
Stress Field Calculation for Quasi-static Ultrasound
Elastography via Force Sensor Integration and SLE Lili Yuan, Ultrasound Laboratory, ECE Department
Advisor: Professor Peder C. Pedersen
Abstract—Most current elastography methods remain qualitative
or display only strain information due to the inexact internal stress
distribution. A more accurate result may be obtained by employing
force sensors to gauge the applied vertical compression force on
skin surface through the ultrasound transducer. A computationally
efficient superposition algorithm based on Love’s closed-form
equation (SLE) is presented. It performs analytical calculation of
the 3D dynamic stress field inside a soft tissue phantom with non-
free boundary conditions under a non-uniformly stressed
rectangular linear array transducer, using the pressure on each
element of the contact surface. The validity of the SLE method was
tested by comparison to the stress field calculated with Finite
Element Analysis (FEA). For the region of interest (directly
underneath the ultrasound transducer), the SLE provides a
sufficiently precise solution and can be exploited for absolute
Young’s modulus reconstruction in real time.
I. INTRODUCTION
Ultrasound elastography is a recent technology for non-
invasively imaging diseased arteries, detecting prostate tumors
and categorizing breast lesions [1]. Well developed quasi-static
free-hand techniques mostly involve only tissue strain. However,
the stiffness may be dependent on the magnitude and direction
of the force applied, especially if the linear stress-strain region
(typically 1%-2%) is exceeded [2]. Inner stress as the response
of external force is also required in addition to the strain.
Inexpensive, ultra-thin, flexible piezoresistive force sensors
(Tekscan, MA) were utilized to quantitatively measure the
contact force. The force sensor output was then integrated into
the SonixRP (Ultrasonix, Vancouver, Canada) strain imaging
system. 3D stress field was achieved by means of the computed
pressure on each force plate element and SLE algorithm.
II. FORCE SENSOR INTEGRATION
A force plate with a slot for the ultrasound transducer was
designed, as shown in Fig. 1. Force sensors were mounted
beneath the force plate and connected to the signal amplification,
noise filtering, and data acquisition system. The force plate was
decomposed into small rectangular elements with each
individual constant pressure. Contact pressure on every sub-
region with a sensor was estimated by measured force divided
by sensing area. For the remaining sub-regions without sensors,
pressure was obtained by bilinear interpolation.
Force Sensors
Force Plate
Transducer Arrayx
y (0,0)
l
w
b
a
Force Plate Element
Fig. 1. Bottom view of linear array rectangular transducer with size of 60 mm ×
20mm, force plate geometry a × b, partitioned into small element with l × w.
A combined display of strain imaging and forcing function
was implemented. Figure 2 illustrates the outcome with agar-
based phantom. Force versus time was used to depict external
force magnitude and loading frequency.
Container filled with Water
PhantomSofter Cyst
SonixRPLinear Array Transducer
Force Sensors
Data Acquisition
System
Force Plate
Strain Image overlaid on B-mode
Forcing Function
Vertical Force
Fig. 2. Front view of sensor integration into strain image system, the upper
image is strain image superimposed on B-mode with blue and red color
respectively representing softer and harder material.
III. SUPERPOSITION ALGORITHM BASED ON LOVE’S FORMULA
SLE was introduced for analytical computation of 3D stress
field under the assumption of a homogeneous, isotropic, semi-
infinite, elastic solid material, stressed non-uniformly by
rectangular compressor. The stress at each field point is the
overall contribution from every force plate element under
individual constant pressure.
node A
z
yx
uniform compression
force plate th element
cross-section of 3D matrix
in the same plane of
transducer beam
1a
4d 3c
2b
0( )' ', ,x y
r
l
w
( )P ti
( , , 0)m ni i
( , , )x y z
( , , 0)2 2
l wm ni i ( , , 0)
2 2
l wm ni i
( , , 0)2 2
l wm ni i ( , , 0)
2 2
l wm ni i
i
field point
node Cnode D
node B
Fig. 3. Geometry of the 𝑖th force element with center and nodes coordinates
Expression for the distribution of vertical stress component
𝑆𝑧𝑧 at field point (𝑥, 𝑦, 𝑧) is
𝑆𝑧𝑧 𝑥, 𝑦, 𝑧 = 𝑆𝑧𝑧𝑖 𝑥, 𝑦, 𝑧
𝑥𝑛𝑜 ∗𝑦𝑛𝑜
𝑖=1 = 1
2𝜋[𝜕𝑉
𝜕𝑧− 𝑧
𝜕2𝑉
𝜕𝑧 2]
𝑥𝑛𝑜 ∗𝑦𝑛𝑜
𝑖=1 (1)
where, 𝑥𝑛𝑜 and 𝑦𝑛𝑜 denote the number of force elements in 𝑥 and
𝑦 direction, 𝑉 is the Newtonian potential, and 𝜕𝑉
𝜕𝑧,
𝜕2𝑉
𝜕𝑧 2 are
calculated by replacing contact area geometry in corresponding
equations as 𝑥𝑑 = 𝑚𝑖 −𝑙
2, 𝑥𝑢 = 𝑚𝑖 +
𝑙
2, 𝑦𝑑 = 𝑛𝑖 −
𝑤
2, 𝑦𝑢 = 𝑛𝑖 +
𝑤
2.
Elaborate description of these formulas omitted here for saving
space can be found in [3].
IV. RESULTS
A phantom (120mm × 60mm × 40mm , Young’s modulus
5 × 104 𝑃𝑎 , Poisson’s ratio 0.42 , density 1040 𝑘𝑔/𝑚3 ) and
2
coaxial force plate ( 108mm × 34mm ) were simulated for 3D
stress distribution calculation using both SLE and FEA
(COMSOL, Sweden).
Cross-sections coincident with transducer image plane from
the two methods were compared to evaluate the accuracy of SLE
for stress estimation in visco-elastic material with fixed
boundary.
A. Stress Field Calculation by SLE
Stress distribution, due to uniform pressure 1000 Pa, along the
axis of compressor decays with depth in a semi-infinite medium,
presented in Fig. 4. The counterpart by FEA with the same
tendency was described in [4].
Fig. 4. Stress field cross section due to uniform vertical compression by SLE.
As an initial experiment, two force sensors were allocated at
two ends of the force plate and the contact surface were divided
into three equal-size rectangle, where the pressure are 1000 𝑃𝑎,
1200 𝑃𝑎, and 1400 𝑃𝑎 in turn along 𝑥 axis. Uneven distributed
stress closer to surface and gradual reduction with depth is
shown in Fig. 5.
Fig. 5. Stress field cross-section for non-uniform normal compression by SLE
B. SLE Performance Evaluation
Stress results from FEA with fixed bottom and prescribed-
displacement surrounding boundary (only displacement in z
direction is allowed) is presented in Fig. 6, followed by the
normalized difference image (Fig. 7). SLE is far more efficient
than FEA (9.2 × 10−6 𝑠/𝑒𝑙𝑒𝑚𝑒𝑛𝑡 vs 1.07 × 10−2 𝑠/𝑒𝑙𝑒𝑚𝑒𝑛𝑡 ) and
less than 5.62% deviation from FEA from surface to depth at
22.3 mm except around the edges.
Fig. 6. 2D stress for non-uniform normal compression using FEA
Fig. 7. Normalized stress difference between SLE and FEA for variable pressure
The mismatch that exists because of gradient of pressure
variation, plate and phantom geometry, mesh fineness and
interpolation method for FEA simulation is not mentioned in
detail due to space limitation.
V. CONCLUSION
By virtue of force sensors and SLE, fast 3D dynamic stress
field calculations are achievable, which can be combined with
strain to obtain real time quantitative elastography for elastic
solid material, therefore improving the quality of tumor and
cancer diagnosis.
ACKNOWLEDGEMENT
The support from the Telemedicine and Advanced
Technology Research Center are greatly appreciated. The
authors also thank research IT support Siamak Najafi for
providing COMSOL software for free and thank lab manager
Frederick Huston for supplying sensor calibration equipments.
REFERENCES
[1] Hall TJ, Zhu Y, Spalding CS, “In vivo real-time freehand palpation
imaging”. Ultrasound Med Biol, 29 (3), 2003, pp. 427-435.
[2] Krouskop, T. A., Wheeler, T. M., kallel, F., Garra, B. S., and Hall, T.,
“Elastic moduli of breast and prostate tissues under compression”,
Ultrasonic Imaging, Vol. 2, 1998, pp. 260-274.
[2] Love, A. E. H, “The Stress Produced in a Semi-infinite Solid by Pressure
on Part of the Boundary,” Phil. Trans. Roy. Soc. London, vol. A228, 1929,
pp. 377-420.
[3] Lili Yuan, Peder C. Pedersen, “Stress Field Simulation for Quantitive
Ultrasound Elasticity Imaging,” Proceedings of the COMSOL Conference,
Boston, October 2009.
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